Justification of the Nonlinear Schrödinger Equation for the Evolution of Gravity Driven 2D Surface Water Waves in a Canal of Finite Depth

Düll, Wolf-Patrick; Schneider, Guido; Wayne, C. Eugene

doi:10.1007/s00205-015-0937-z

Cited by 35 publications

(46 citation statements)

References 40 publications

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“…The proof of Lemma 2.1 is analogous to that of Lemma 2.6 in [9] (see also [6]). In fact, the first and the third estimates are valid for appropriate constants C Res and C Ψ for all s > 0, this is a consequence of the fact that our approximation ǫΨ has compact support in Fourier space.…”

Section: Introductionmentioning

confidence: 89%

“…Since ±ω(mk 0 ) = ±mω(k 0 ) for all integers m ≥ 2, we can proceed analogously as in [9] to replace ǫ Ψ by a new approximation ǫΨ of the form…”

Section: Introductionmentioning

confidence: 99%

“…where A −j = A j and A −jℓ = A jℓ have compact support in Fourier space for all 0 < ǫ ≪ 1. Then, exactly as in Section 2 of [9], the following estimates hold for the modified residual.…”

Section: Introductionmentioning

confidence: 99%

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Justification of the NLS Approximation for the Euler–Poisson Equation

Liu

2019

Commun. Math. Phys.

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The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler-Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation. 2000 Mathematics Subject Classification. 35M20; 35Q35. 1 Jang considered the global solution with spherical symmetry initial data [23]. Furthermore, Jang, Li and Zhang obtained the smooth global solutions [24]. Finally, Li and Wu solved the Cauchy problem for the two dimensional electron Euler-Poisson system [30]. Recently, Guo, Han and Zhang [11] finally completely settled this problem of global existence and proved that no shocks form for the 1D Euler-Poisson system for electrons. For the Euler-Poisson equation for ions, Guo and Pausader [13] constructed global smooth irrotational solutions with small amplitude for ion dynamics in R 3 . For the long wave approximation, Guo and Pu [14] established rigorously the KdV limit for the ion Euler-Poisson system in 1D for both cold and hot plasma cases, where the electron density satisfies the classical Maxwell-Boltzmann law. This result was generalized to the higher dimensional cases, and the 2D Kadomtsev-Petviashvili-II (KP-II) equation and the 3D Zakharov-Kuznetsov (ZK) equation are derived under different scalings [31]. Almost at the same time, [27] also established the Zakharov-Kuznetsov equation in 3D from the Euler-Poisson system. Recently, the authors in the present paper [28,29] obtained rigorously the quantum KdV limit in 1D and the KP-I and KP-II equations in 2D for the Euler-Poisson system for cold as well as hot plasma taking quantum effect into account, where the electron equilibrium is given by a Fermi-Dirac distribution. Han-Kwan [15] also introduced a long wave scaling for the Vlasov-Poisson equation

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Section: Introductionmentioning

confidence: 89%

“…Since ±ω(mk 0 ) = ±mω(k 0 ) for all integers m ≥ 2, we can proceed analogously as in [9] to replace ǫ Ψ by a new approximation ǫΨ of the form…”

Section: Introductionmentioning

confidence: 99%

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Justification of the NLS Approximation for the Euler–Poisson Equation

Liu

2019

Commun. Math. Phys.

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“…On the basis of the form of the ansatz one expects an approximation result to hold for times O( −2 ). While [14] and subsequently [5] gave a result that held on the correct qualitative time interval, the result was linked to a specific time T 1 / 2 . One would like an arbitrary time T 0 / 2 with T 0 coming from the approximation Ψ N LS .…”

Section: Introductionmentioning

confidence: 94%

Modified energy functionals and the NLS approximation

Cummings¹,

Wayne²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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We consider a model equation from [14] that captures important properties of the water wave equation. We give a new proof of the fact that wave packet solutions of this equation are approximated by the nonlinear Schrödinger equation. This proof both simplifies and strengthens the results of [14] so that the approximation holds for the full interval of existence of the approximate NLS solution rather than just a subinterval. Furthermore, the proof avoids the problems associated with inverting the normal form transform in [14] by working with a modified energy functional motivated by [1] and [8].

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“…The (resonant) four wave interaction (FWI) system has been justified for a situation as it appears for gravity surface water waves. In principle existing approximation results for the NLS approximation, especially for the water wave problem [8,9] can be carried over line for line to the FWI approximation and so the FWI approximation can be expected to be valid for the water wave problem in case of pure gravity surface water waves in finite and infinite depth.…”

Section: Introductionmentioning

confidence: 99%